Learn Dutch: Ik voel me 's morgens altijd een beetje grieperig en slaperig. - I always feel a bit flu-like and sleepy in the morning.

The suffix -erig often indicates a mild state or a tendency. Adding it to the noun griep (flu) creates grieperig, which means 'flu-like' or 'under the weather.' You aren't necessarily diagnosed with the full-blown flu, but you are experiencing mild symptoms resembling it.

Why use 'slaperig' instead of a present participle like 'slapend'?

The suffix -erig attached to the root slaap creates an adjective describing a feeling or behavioral tendency, so slaperig means 'sleepy' or 'drowsy.' If you used the participle slapend, it would mean you are literally 'sleeping' at that moment, which contradicts the fact that you are awake and noticing how you feel.

Is 'een beetje' redundant since words ending in '-erig' already imply a mild state?

It is not strictly necessary, but it is definitely not considered redundant in everyday Dutch. Speakers very often pair een beetje with -erig adjectives to soften the statement even further and emphasize that the feeling is just a slight, lingering sensation.

Could I drop 'me' and just say 'Ik voel 's morgens altijd...'?

No, you need me here because the verb zich voelen (to feel an emotion or internal state) is reflexive in Dutch. If you drop the reflexive pronoun me, the verb voelen just means 'to physically touch' or 'to feel an object with your hands.'